<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">M1</font> be    <a href="pboole.html#M1">ManySortedSet</a> of  <a href="hilbert1.html#K1">HP-WFF</a> , <font color="Maroon">M2</font> be    <a href="pboole.html#M1">ManySortedSet</a> of  <a href="hilbert1.html#K1">HP-WFF</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:41_1_2"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">M1</font> <a href="funct_1.html#K1">.</a> <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K2">root-tree</a> <a href="hilbert1.html#K2">VERUM</a> 
 <b>and </b><br/><a NAME="E2:41_1_2"/><i><font color="Green">E92</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">M1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K2">root-tree</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span>
 <b>and </b><br/><a NAME="E3:41_1_2"/><i><font color="Green">E93</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   ex <font color="Olive">p'</font>, <font color="Olive">q'</font> being  <a href="trees_2.html#NM6">DecoratedTree</a> of  <a href="hilbert1.html#K1">HP-WFF</a>  st <br/>( <font color="Olive">p'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">q'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Maroon">M1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font></span>)</span> <a href="trees_4.html#K10">-tree</a> <font color="Olive">p'</font>,<font color="Olive">q'</font> &amp; <font color="Maroon">M1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font></span>)</span> <a href="trees_4.html#K10">-tree</a> <font color="Olive">p'</font>,<font color="Olive">q'</font> )
 <b>and </b><br/><a NAME="E4:41_1_2"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">M2</font> <a href="funct_1.html#K1">.</a> <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K2">root-tree</a> <a href="hilbert1.html#K2">VERUM</a> 
 <b>and </b><br/><a NAME="E5:41_1_2"/><i><font color="Green">E95</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">M2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K2">root-tree</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span>
 <b>and </b><br/><a NAME="E6:41_1_2"/><i><font color="Green">E96</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   ex <font color="Olive">p'</font>, <font color="Olive">q'</font> being  <a href="trees_2.html#NM6">DecoratedTree</a> of  <a href="hilbert1.html#K1">HP-WFF</a>  st <br/>( <font color="Olive">p'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">q'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Maroon">M2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font></span>)</span> <a href="trees_4.html#K10">-tree</a> <font color="Olive">p'</font>,<font color="Olive">q'</font> &amp; <font color="Maroon">M2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font></span>)</span> <a href="trees_4.html#K10">-tree</a> <font color="Olive">p'</font>,<font color="Olive">q'</font> )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a> ] <b>means </b><font color="Maroon">M1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">M2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E7:41_1_2"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="hilbert1.html#K2">VERUM</a> ]
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:41_1_2"/><i><font color="Green">E98</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[ <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>thus </b><font color="Maroon">M1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Maroon">n</font></span>)</span> = 
 <a href="trees_4.html#K2">root-tree</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">M2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i/>
;<br/>


</div><b>end;</b></div>
<a NAME="E9:41_1_2"/><i><font color="Green">E99</font></i>: 
for <font color="Olive">r</font>, <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">r</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">s</font>] holds <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">r</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">s</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">r</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">s</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a> ;<br/><b>let </b><font color="Maroon">s</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a> ;<br/>



<b>assume </b><a NAME="E1:41_1_2_2"/><i><font color="Green">E100</font></i>: 
( <font color="Maroon">M1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font> &amp; <font color="Maroon">M1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> )
 ;<br/>

<b>consider </b><font color="Maroon">p'</font> being   <a href="trees_2.html#NM6">DecoratedTree</a> of  <a href="hilbert1.html#K1">HP-WFF</a> , <font color="Maroon">q'</font> being   <a href="trees_2.html#NM6">DecoratedTree</a> of  <a href="hilbert1.html#K1">HP-WFF</a> <b> such that </b><br/><a NAME="E3:41_1_2_2"/><i><font color="Green">E125</font></i>: 
( <font color="Maroon">p'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font> &amp; <font color="Maroon">q'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> )
 <b>and </b><br/><a NAME="E4:41_1_2_2"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">M1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">s</font></span>)</span> <a href="trees_4.html#K10">-tree</a> <font color="Maroon">p'</font>,<font color="Maroon">q'</font>
 <b>and </b><br/><a NAME="E5:41_1_2_2"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">M1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">s</font></span>)</span> <a href="trees_4.html#K10">-tree</a> <font color="Maroon">p'</font>,<font color="Maroon">q'</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">p'</font> being   <a href="trees_2.html#NM6">DecoratedTree</a> of  <a href="hilbert1.html#K1">HP-WFF</a> , <font color="Maroon">q'</font> being   <a href="trees_2.html#NM6">DecoratedTree</a> of  <a href="hilbert1.html#K1">HP-WFF</a> <b> such that </b><br/><a NAME="E7:41_1_2_2"/><i><font color="Green">E128</font></i>: 
( <font color="Maroon">p'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font> &amp; <font color="Maroon">q'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> )
 <b>and </b><br/><a NAME="E8:41_1_2_2"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">M2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">s</font></span>)</span> <a href="trees_4.html#K10">-tree</a> <font color="Maroon">p'</font>,<font color="Maroon">q'</font>
 <b>and </b><br/><a NAME="E9:41_1_2_2"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">M2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">s</font></span>)</span> <a href="trees_4.html#K10">-tree</a> <font color="Maroon">p'</font>,<font color="Maroon">q'</font>
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E10:41_1_2_2"/>
<font color="Maroon">M1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">M2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">s</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="hilbert2.html#T2" target="_self">Th2</a>, <a class="ref" href="hilbert2.html#T3" target="_self">Th3</a>, <a class="ref" href="hilbert2.html#T4" target="_self">Th4</a></i>;<br/>

<b>thus </b><a NAME="E11:41_1_2_2"/>
<font color="Maroon">M1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">M2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">s</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="hilbert2.html#T3" target="_self">Th3</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:41_1_2"/>
for <font color="Olive">r</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">r</font>]
 
<b>from </b><i><a class="ref" href="hilbert2.html#S2" target="_self">HILBERT2:sch 2</a>(, , );<br/></i>
<a NAME="E11:41_1_2"/><b>then </b>
for <font color="Olive">r</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="hilbert1.html#K1">HP-WFF</a>  holds <br/><font color="Maroon">M1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">r</font>
 
;<br/>
<b>hence </b><a NAME="E12:41_1_2"/>
<font color="Maroon">M1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">M2</font>
 <b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


</div>
